I have looked at the example – and there is major difference in those examples vs. a regular WordPress powered page, and that it's not using wordpress at all. While WordPress provides a great interface to make it easier to build websites, each request that is made, has to be made through WordPress itself – which means that every ajax request has to be sent to the wordpress ajax request handler, which builds up the whole system, loads the plugins, themes etc.. – which takes a lot of time.
Not using WordPress, but a separate search handler built for a purpose gets rid off that overhead, and makes the search process a lot faster, and by a lot I mean a huge amount – as there is no need for building up the whole system, just do the search, and return the results, and it's done. With WordPress this is impossible, as it is no way to access the database, post types, images, shortcodes, nothing in general, without loading the core and all plugins etc… Some plugin providers get around this by offering 3rd party search servers, but those have a monthly subscription fee, which depends on the search volume as well.
I'm working on a possible pre-cache solution that would use the client browser to do the most of the search 'instantly', but that won't be available for a long time yet – as it has many downsides I need to resolve. Unfortunately speed does not come for free now days. This plugin does not need a subscription, it works on the server that is installed on, however it depends on that server performance. Other solutions may offer 3rd party search servers, but that of course comes with a subscription fee, usually from 15$/month and above, which is half the price of this product, for a lifetime.
There are a few very effective tips to improve the search performance though, which you can find here: Fine tuning the search configuration – Documentation
Beyond that, it depends only on the server performance.
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